scholarly journals The distribution of the number of small cuts in a random planar triangulation

2010 ◽  
Vol DMTCS Proceedings vol. AM,... (Proceedings) ◽  
Author(s):  
Zhicheng Gao ◽  
Gilles Schaeffer
Keyword(s):  
Connected Components ◽  
Planar Triangulation ◽  
Asymptotically Normal ◽  
International Audience

International audience We enumerate rooted 3-connected (2-connected) planar triangulations with respect to the vertices and 3-cuts (2-cuts). Consequently, we show that the distribution of the number of 3-cuts in a random rooted 3-connected planar triangulation with $n+3$ vertices is asymptotically normal with mean $(10/27)n$ and variance $(320/729)n$, and the distribution of the number of 2-cuts in a random 2-connected planar triangulation with $n+2$ vertices is asymptotically normal with mean $(8/27)n$ and variance $(152/729)n$. We also show that the distribution of the number of 3-connected components in a random 2-connected triangulation with $n+2$ vertices is asymptotically normal with mean $n/3$ and variance $\frac{8}{ 27}n$ .

scholarly journals Sorting using complete subintervals and the maximum number of runs in a randomly evolving sequence: Extended abstract.

2007 ◽  
Vol DMTCS Proceedings vol. AH,... (Proceedings) ◽  
Author(s):  
Svante Janson
Keyword(s):  
Order Term ◽  
Combinatorial Problem ◽  
Random Order ◽  
Priority Queues ◽  
Sorting Algorithm ◽  
Asymptotic Results ◽  
First Order ◽  
Asymptotically Normal ◽  
International Audience ◽  
Space Requirements

International audience We study the space requirements of a sorting algorithm where only items that at the end will be adjacent are kept together. This is equivalent to the following combinatorial problem: Consider a string of fixed length n that starts as a string of 0's, and then evolves by changing each 0 to 1, with the n changes done in random order. What is the maximal number of runs of 1's? We give asymptotic results for the distribution and mean. It turns out that, as in many problems involving a maximum, the maximum is asymptotically normal, with fluctuations of order $n^{1/2}$, and to the first order well approximated by the number of runs at the instance when the expectation is maximized, in this case when half the elements have changed to 1; there is also a second order term of order $n^{1/3}$. We also treat some variations, including priority queues and sock-sorting.

scholarly journals On the bialgebra of functional graphs and differential algebras

1997 ◽  
Vol Vol. 1 ◽  
Author(s):  
Maurice Ginocchio
Keyword(s):  
Dynamical Systems ◽  
Differential Operators ◽  
Discrete Dynamical Systems ◽  
Rooted Trees ◽  
Connected Components ◽  
Differential Algebras ◽  
Noncommutative Polynomials ◽  
International Audience

International audience We develop the bialgebraic structure based on the set of functional graphs, which generalize the case of the forests of rooted trees. We use noncommutative polynomials as generating monomials of the functional graphs, and we introduce circular and arborescent brackets in accordance with the decomposition in connected components of the graph of a mapping of \1, 2, \ldots, n\ in itself as in the frame of the discrete dynamical systems. We give applications fordifferential algebras and algebras of differential operators.

scholarly journals Toward the asymptotic count of bi-modular hidden patterns under probabilistic dynamical sources: a case study

2012 ◽  
Vol DMTCS Proceedings vol. AQ,... (Proceedings) ◽  
Author(s):  
Loïck Lhote ◽  
Manuel E. Lladser
Keyword(s):  
Technical Difficulty ◽  
Memory Length ◽  
Asymptotically Normal ◽  
International Audience ◽  
Finite Memory ◽  
Hidden Patterns ◽  
Context Free ◽  
With Memory ◽  
Random Text

International audience Consider a countable alphabet $\mathcal{A}$. A multi-modular hidden pattern is an $r$-tuple $(w_1,\ldots , w_r)$, where each $w_i$ is a word over $\mathcal{A}$ called a module. The hidden pattern is said to occur in a text $t$ when the later admits the decomposition $t = v_0 w_1v_1 \cdots v_{r−1}w_r v_r$, for arbitrary words $v_i$ over $\mathcal{A}$. Flajolet, Szpankowski and Vallée (2006) proved via the method of moments that the number of matches (or occurrences) with a multi-modular hidden pattern in a random text $X_1\cdots X_n$ is asymptotically Normal, when $(X_n)_{n\geq1}$ are independent and identically distributed $\mathcal{A}$-valued random variables. Bourdon and Vallée (2002) had conjectured however that asymptotic Normality holds more generally when $(X_n)_{n\geq1}$ is produced by an expansive dynamical source. Whereas memoryless and Markovian sequences are instances of dynamical sources with finite memory length, general dynamical sources may be non-Markovian i.e. convey an infinite memory length. The technical difficulty to count hidden patterns under sources with memory is the context-free nature of these patterns as well as the lack of logarithm-and exponential-type transformations to rewrite the product of non-commuting transfer operators. In this paper, we address a case study in which we have successfully overpassed the aforementioned difficulties and which may illuminate how to address more general cases via auto-correlation operators. Our main result shows that the number of matches with a bi-modular pattern $(w_1, w_2)$ normalized by the number of matches with the pattern $w_1$, where $w_1$ and $w_2$ are different alphabet characters, is indeed asymptotically Normal when $(X_n)_{n\geq1}$ is produced by a holomorphic probabilistic dynamical source.

scholarly journals Percolation on a non-homogeneous Poisson blob process

2003 ◽  
Vol DMTCS Proceedings vol. AC,... (Proceedings) ◽  
Author(s):  
Fabio P. Machado
Keyword(s):  
Connected Components ◽  
Multi Scale ◽  
International Audience

International audience We present the main results of a study for the existence of vacant and occupied unbounded connected components in a non-homogeneous Poisson blob process. The method used in the proofs is a multi-scale percolation comparison.

scholarly journals Random 2-SAT Solution Components and a Fitness Landscape

2011 ◽  
Vol Vol. 13 no. 2 (Graph and Algorithms) ◽  
Author(s):  
Damien Pitman
Keyword(s):  
Fitness Landscape ◽  
Biological Evolution ◽  
Random Variable ◽  
Single Site ◽  
Connected Components ◽  
Number Of Components ◽  
Bounded Number ◽  
International Audience ◽  
Stochastically Bounded ◽  
Probability Range

Graphs and Algorithms International audience We describe a limiting distribution for the number of connected components in the subgraph of the discrete cube induced by the satisfying assignments to a random 2-SAT formula. We show that, for the probability range where formulas are likely to be satisfied, the random number of components converges weakly (in the number of variables) to a distribution determined by a Poisson random variable. The number of satisfying assignments or solutions is known to grow exponentially in the number of variables. Thus, our result implies that exponentially many solutions are organized into a stochastically bounded number of components. We also describe an application to biological evolution; in particular, to a type of fitness landscape where satisfying assignments represent viable genotypes and connectivity of genotypes is limited by single site mutations. The biological result is that, with probability approaching 1, each viable genotype is connected by single site mutations to an exponential number of other viable genotypes while the number of viable clusters is finite.

scholarly journals Decomposable graphs and definitions with no quantifier alternation

2005 ◽  
Vol DMTCS Proceedings vol. AE,... (Proceedings) ◽  
Author(s):  
Oleg Pikhurko ◽  
Joel Spencer ◽  
Oleg Verbitsky
Keyword(s):  
The Other ◽  
Connected Components ◽  
Large Graphs ◽  
First Order ◽  
Minimum Number ◽  
International Audience ◽  
Decomposable Graphs ◽  
Number Of Iterations ◽  
Quantifier Alternation ◽  
Binary Logarithm

International audience Let $D(G)$ be the minimum quantifier depth of a first order sentence $\Phi$ that defines a graph $G$ up to isomorphism in terms of the adjacency and the equality relations. Let $D_0(G)$ be a variant of $D(G)$ where we do not allow quantifier alternations in $\Phi$. Using large graphs decomposable in complement-connected components by a short sequence of serial and parallel decompositions, we show examples of $G$ on $n$ vertices with $D_0(G) \leq 2 \log^{\ast}n+O(1)$. On the other hand, we prove a lower bound $D_0(G) \geq \log^{\ast}n-\log^{\ast}\log^{\ast}n-O(1)$ for all $G$. Here $\log^{\ast}n$ is equal to the minimum number of iterations of the binary logarithm needed to bring $n$ below $1$.

scholarly journals A simple formula for bipartite and quasi-bipartite maps with boundaries

2012 ◽  
Vol DMTCS Proceedings vol. AR,... (Proceedings) ◽  
Author(s):  
Gwendal Collet ◽  
Eric Fusy
Keyword(s):  
Generating Function ◽  
Simple Formula ◽  
Connected Components ◽  
Nous Obtenons ◽  
Single Tree ◽  
International Audience ◽  
Planar Maps

International audience We obtain a very simple formula for the generating function of bipartite (resp. quasi-bipartite) planar maps with boundaries (holes) of prescribed lengths, which generalizes certain expressions obtained by Eynard in a book to appear. The formula is derived from a bijection due to Bouttier, Di Francesco and Guitter combined with a process (reminiscent of a construction of Pitman) of aggregating connected components of a forest into a single tree. Nous obtenons une formule très simple pour la série génératrice des cartes biparties ayant des bords (trous) de tailles fixées, généralisant certaines expressions obtenues par Eynard dans un livre à paraître. Nous obtenons la formule à partir d'une bijection due à Bouttier, Di Francesco et Guitter, combinée avec un processus (dans l'esprit d'une construction due à Pitman) pour agréger les composantes connexes d'une forêt en un unique arbre.

scholarly journals Coloring Rings in Species

2014 ◽  
Vol DMTCS Proceedings vol. AT,... (Proceedings) ◽  
Author(s):  
Jacob White
Keyword(s):  
Directed Graph ◽  
Symmetric Function ◽  
Chromatic Polynomial ◽  
Connected Components ◽  
Expansion Formula ◽  
Strongly Connected Components ◽  
Set Partitions ◽  
Strongly Connected ◽  
International Audience ◽  
Chromatic Symmetric Function

International audience We present a generalization of the chromatic polynomial, and chromatic symmetric function, arising in the study of combinatorial species. These invariants are defined for modules over lattice rings in species. The primary examples are graphs and set partitions. For these new invariants, we present analogues of results regarding stable partitions, the bond lattice, the deletion-contraction recurrence, and the subset expansion formula. We also present two detailed examples, one related to enumerating subgraphs by their blocks, and a second example related to enumerating subgraphs of a directed graph by their strongly connected components.

scholarly journals Mixing times of Markov chains on 3-Orientations of Planar Triangulations

2012 ◽  
Vol DMTCS Proceedings vol. AQ,... (Proceedings) ◽  
Author(s):  
Sarah Miracle ◽  
Dana Randall ◽  
Amanda Pascoe Streib ◽  
Prasad Tetali
Keyword(s):  
Markov Chain ◽  
Markov Chains ◽  
State Space ◽  
Maximum Degree ◽  
Edge Coloring ◽  
Fixed Number ◽  
Mixing Times ◽  
Planar Triangulation ◽  
International Audience ◽  
High Degree

International audience Given a planar triangulation, a 3-orientation is an orientation of the internal edges so all internal vertices have out-degree three. Each 3-orientation gives rise to a unique edge coloring known as a $\textit{Schnyder wood}$ that has proven useful for various computing and combinatorics applications. We consider natural Markov chains for sampling uniformly from the set of 3-orientations. First, we study a "triangle-reversing'' chain on the space of 3-orientations of a fixed triangulation that reverses the orientation of the edges around a triangle in each move. We show that (i) when restricted to planar triangulations of maximum degree six, the Markov chain is rapidly mixing, and (ii) there exists a triangulation with high degree on which this Markov chain mixes slowly. Next, we consider an "edge-flipping'' chain on the larger state space consisting of 3-orientations of all planar triangulations on a fixed number of vertices. It was also shown previously that this chain connects the state space and we prove that the chain is always rapidly mixing.

scholarly journals New combinatorial computational methods arising from pseudo-singletons

2008 ◽  
Vol DMTCS Proceedings vol. AJ,... (Proceedings) ◽  
Author(s):  
Gilbert Labelle
Keyword(s):  
Power Series ◽  
Nous Avons ◽  
Computational Methods ◽  
Exponential Function ◽  
Connected Components ◽  
Finite Sets ◽  
Connected Sets ◽  
International Audience ◽  
Analytical Power ◽  
Nous Utilisons

International audience Since singletons are the connected sets, the species $X$ of singletons can be considered as the combinatorial logarithm of the species $E(X)$ of finite sets. In a previous work, we introduced the (rational) species $\widehat{X}$ of pseudo-singletons as the analytical logarithm of the species of finite sets. It follows that $E(X) = \exp (\widehat{X})$ in the context of rational species, where $\exp (T)$ denotes the classical analytical power series for the exponential function in the variable $T$. In the present work, we use the species $\widehat{X}$ to create new efficient recursive schemes for the computation of molecular expansions of species of rooted trees, of species of assemblies of structures, of the combinatorial logarithm species, of species of connected structures, and of species of structures with weighted connected components. Puisque les singletons sont les ensembles connexes, l'espèce $X$ des singletons peut être considérée comme le logarithme combinatoire de l'espèce $E(X)$ des ensembles finis. Dans un travail antérieur, nous avons introduit l'espèce (rationnelle) $\widehat{X}$ des pseudo-singletons comme étant le logarithme analytique de l'espèce des ensembles finis. Il en découle que $E(X) = \exp (\widehat{X})$ dans le contexte des espèces rationnelles, où $\exp (T)$ désigne la série de puissances analytique classique de la fonction exponentielle dans la variable $T$. Dans le présent travail, nous utilisons l'espèce $\widehat{X}$ pour créer de nouveaux schémas computationnels récursifs efficaces pour le calcul du développement moléculaire de l'espèce des arborescences, d'espèces d'assemblées de structures, de l'espèce du logarithme combinatoire, d'espèces de structures connexes, et d'espèces de structures à composantes connexes pondérées.

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